Non-Dimensional Equation of Motion of a Two-Dimensional Beam Using Stiffness Matrices Based on Structural Mechanics in the Absolute Nodal Coordinate Formulation

2020 ◽  
Vol 44 (8) ◽  
pp. 581-588
Author(s):  
Ji Heon Kang ◽  
Jae Wook Lee ◽  
Jin Seok Jang ◽  
Chang Young Choi ◽  
Kun Woo Kim
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Structural Mechanics ◽  
Equation Of Motion ◽  
Two Dimensional ◽  
Absolute Nodal Coordinate ◽  
Stiffness Matrices ◽  
Dimensional Equation ◽  
The Absolute

Nondimensional analysis of a two-dimensional shear deformable beam in absolute nodal coordinate formulation

2017 ◽  
Vol 232 (7) ◽  
pp. 1236-1246 ◽  
Author(s):  
Kun-Woo Kim ◽  
Jae-Wook Lee ◽  
Jin-Seok Jang ◽  
Joo-Young Oh ◽  
Ji-Heon Kang ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Absolute Nodal Coordinate Formulation ◽  
Equation Of Motion ◽  
Central Processing Unit ◽  
Processing Unit ◽  
Two Dimensional ◽  
Absolute Nodal Coordinate ◽  
Central Processing ◽  
Dimensional Equation ◽  
Shear Deformable

Absolute nodal-coordinate formulation is a technique that was developed in 1996 for expressing the large rotation and deformation of a flexible body. It utilizes global slopes without a finite rotation in order to define nodal coordinates. The method has a shortcoming in that the central processing unit time increases because of increases in the degrees of freedom. In particular, when considering the deformation of a cross section, the shortcoming due to the increase in the degrees of freedom becomes clear. Therefore, in the present research, the dimensional equation of motion concerning a two-dimensional shear deformable beam, developed by Omar and Shabana, is converted into a nondimensional equation of motion in order to reduce the central processing unit time. By utilizing an example of a cantilever beam, wherein an exact solution for the static deflection exists, the nondimensional equation of motion was verified. Moreover, by using an example of a free-falling flexible pendulum, the efficiency of the nondimensional equation of motion gained by increasing the number of elements was compared with that of the dimensional equation of motion.

A Linear Beam Finite Element Based on the Absolute Nodal Coordinate Formulation

2004 ◽  
Vol 127 (4) ◽  
pp. 621-630 ◽  
Author(s):  
Kimmo S. Kerkkänen ◽  
Jussi T. Sopanen ◽  
Aki M. Mikkola
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Longitudinal Direction ◽  
Beam Element ◽  
Computational Effort ◽  
Two Dimensional ◽  
Absolute Nodal Coordinate ◽  
Beam Elements ◽  
Shear Deformable Beam ◽  
The Absolute ◽  
Shear Deformable

In this paper, a new two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation is proposed. The nonlinear elastic forces of the beam element are obtained using a continuum mechanics approach, without employing a local element coordinate system. In this study, linear polynomials are used to interpolate both the transverse and longitudinal components of the displacement. This is different from other absolute nodal-coordinate-based beam elements where cubic polynomials are used in the longitudinal direction. The use of linear interpolation polynomials leads to the phenomenon known as shear locking. This defect is avoided through the adoption of selective integration within the numerical integration method. The proposed element is verified using several numerical examples. The results of the proposed element are compared to analytical solutions and the results for an existing shear deformable beam element. It is shown that by using the proposed element, accurate linear and nonlinear static deformations, as well as realistic dynamic behavior including the capturing of the centrifugal stiffening effect, can be achieved with a smaller computational effort than by using existing shear deformable two-dimensional beam elements.

Non-Linear Strain Description for Two-Dimensional Shear Deformable Beam Element Based on the Absolute Nodal Coordinate Formulation

2009 ◽  
Author(s):  
Jussi Sopanen ◽  
Marko Matikainen ◽  
Aki Mikkola
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Two Dimensional ◽  
Linear Strain ◽  
Absolute Nodal Coordinate ◽  
Elastic Forces ◽  
Longitudinal Slope ◽  
The Absolute

In the absolute nodal coordinate formulation, the transverse shear deformation can be accounted for by using a fully parametrized element or, alternatively, by replacing longitudinal slope coordinates by a vector that describes the orientation of the cross-section. The use of a fully parametrized element allows the description of cross-section deformations in the case of beams and, correspondingly, fiber deformations in the case of plates and shells. It is noteworthy, however, that cross-section or fiber deformations are usually associated with high natural frequencies complicating the time integration of a fully parametrized element. A procedure to replace longitudinal slope coordinates by the vector that describes cross-section orientation was recently applied to a two-dimensional beam element based on the absolute nodal coordinate formulation. In this study, the procedure to account for shear deformation using the vector that describes cross-section orientation is extended to account for the nonlinear strain-displacement relationship in the definition of the elastic forces of the beam element. To accomplish this, the exact displacement field is used in the description of element kinematical and strain measures. This makes it straightforward to implement the non-linear strain-displacement relationship in the description of the elastic forces. Numerical results demonstrate that the enhanced beam element yields accurate results in eigenfrequency analysis. Results obtained in large deformation cases are in line with previously proposed elements based on the absolute nodal coordinate formulation.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

1999 ◽  
Vol 122 (4) ◽  
pp. 498-507 ◽  
Author(s):  
Marcello Campanelli ◽  
Marcello Berzeri ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Systems ◽  
Large Displacement ◽  
Body Motion ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Finite Element Formulations

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

Elastic Forces and Coordinate Systems in the Finite Element Formulations

1999 ◽  
Author(s):  
Marcello Berzeri ◽  
Marcello Campanelli ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Coordinate Systems ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Elastic Forces ◽  
Floating Frame ◽  
The Absolute ◽  
Finite Element Formulations

Abstract The equivalence of the elastic forces of finite element formulations used in flexible multibody dynamics is the focus of this investigation. Two conceptually different finite element formulations that lead to exact modeling of the rigid body dynamics will be used. These are the floating frame of reference formulation and the absolute nodal coordinate formulation. It is demonstrated in this study that different element coordinate systems, which are used for the convenience of describing the element deformations in the absolute nodal coordinate formulation, lead to similar results as the element size is reduced. The equivalence of the elastic forces in the absolute nodal coordinate and the floating frame of reference formulations is shown. The result of this analysis clearly demonstrates that the instability observed in high speed rotor analytical models due to the neglect of the geometric centrifugal stiffening is not a problem inherent to a particular finite element formulation but only depends on the beam model that is used. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. A new method is presented and used to obtain a simple expression for the elastic forces in the absolute nodal coordinate formulation. This method, which employs a nonlinear elastic strain-displacement relationship, does not result in an unstable solution when the angular velocity is increased.

A Numerical Approach to Solving Flexible Multibody Systems Using the Absolute Nodal Coordinate Formulation

1999 ◽  
Author(s):  
R. Y. Yakoub ◽  
A. A. Shabana
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Sparse Matrix ◽  
Mass Matrix ◽  
Numerical Approach ◽  
Absolute Nodal Coordinates ◽  
Absolute Nodal Coordinate ◽  
Velocity Transformation ◽  
Flexible Multibody ◽  
Nodal Coordinates ◽  
The Absolute

Abstract By utilizing the fact that the absolute nodal coordinate formulation leads to a constant mass matrix, a Cholesky decomposition of the mass matrix can be used to obtain a constant velocity transformation matrix. This velocity transformation can be used to express the absolute nodal coordinates in terms of the generalized Cholesky coordinates. In this case, the inertia matrix associated with the Cholesky coordinates is the identity matrix, and therefore, an optimum sparse matrix structure can be obtained for the augmented multibody equations of motions. The implementation of a computer procedure based on the absolute nodal coordinate formulation and Cholesky coordinates is discussed in this paper. A flexible four-bar linkage is presented in this paper in order to demonstrate the use of Cholesky coordinates in the simulation of the small and large deformations in flexible multibody applications. The results obtained from the absolute nodal coordinate formulation are compared to those obtained from the floating frame of reference formulation.

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